Raw video files are huge. For example, an Ultra High Definition (UHD) movie with 120 frames per second (fps), 3840×2160 pixels per frame, 3 colors per pixel, and 16 bits per color, requires bandwidth of:3840*2160*120*3*16=47,775,744,000 Bits per sec≈50 Giga bits per sec,equivalent to about 500 high speed (100 Mbps) fiber channels.
If the movie last for two hours, as usual, it requires storage of:47,775,744,000*7,200≈343,985 Giga bits≈45 Tera bytes,equivalent to about 5,000 regular (5 Gbytes) DVD disks.
Video compression,                “The art of reducing the video size without affecting its visual quality”, is therefore a necessary tool for any applications that deals with video.        
In general, a video consists of several components, such as in the RGB color space or in the YUV color space. However, without loss of generality we consider here only one such component. The generalization to a whole video is discussed in Pat [1].
The lattice of integers, Zn, is the set of n tuples of integers in the real Euclidean space of Rn. A frame can be viewed as a rectangular grid on the lattice Z2, and a video as a cubic grid on Z3. A subset of a lattice, which is itself a lattice, is called a sub-lattice, see Ref [1]. Examples of sub-lattices of Z2 are given in FIG. 1. The two Quincunx sub-lattices are given in unit 110. The white circled points constitute the even sub-lattice and the dark circled points the odd sub-lattice. The four Dyadic sub-lattices are similarly given in unit 120. A dilation matrix is associated with the sub-lattices, see unit 115 for the Quincunx case and unit 125 for the Dyadic case. Note further that the number of possible sub-lattices is determined by the determinant of the corresponding dilation matrix.
Down-sampling refers to the process of extracting a sub-lattice from a given lattice. For example, we show Dyadic down sampling in FIG. 2. The input signal is shown in unit 210. A temporal down sampling is shown in unit 220, and a spatial down sampling in unit 230. A combined temporal and spatial down sampling is shown in unit 240.
A Generic Video Codec, as depicted in FIG. 3, consists of the following:                1. The Encoder:                    The input video is denoted by Y, and the output encoded video by Y′.                        2. The Bit Stream:                    The Bit Stream is the encoded video Y′.            Depending on the application, it is either transmitted or stored on disk.                        3. The Decoder:                    The input to the Decoder is the Bit Stream, and the output decoded video is denoted by Ŷ.                        
See Pat [1] for more details.